Is there a way to get myself and then my mom a permanent visa in Japan?

There is such a thing as a retirement visa in Japan, but you definitely need means to be eligible. If you want to make a career out of ESL, then you want a Masters in TESOL, etc. 

No such thing as a " permanent  visa " in Japan .

If you're American she can only stay 90 days .

And you're getting ahead of yourself . Have you ever been there ?

If not , you will need to .

Most English teachers only last a year or two before they get burned out and quit .

  You often don't get a choice of where , or whom you teach . It could be 5 year old kids , or 45 year old housewives who have nothing better to do .

Teaching English won't be a lifelong job . Over 30 and you might be history .

    Other jobs you will need a job skill a Japanese doesn't have .

Rents are high . And most everything is in Japanese .

What would she do when she got there ?

Suggestions ? Take your mom to Japan for a two week vacation ...

While having a college degree is one key requirement, it does not guaranty you a job. Your degree has to be pertinent to the job position that is being offered.  

At one time, just being a native English speaker could get you a job at a language school, but those days are LONG GONE.  These days, the language schools hiring English teachers are very picky for among other reasons there is a glut of such people in Japan.  Plus, teaching English is not a long term career and it does not pay that much that you can support both your mother and yourself.  You may think that just because your mother will get retirement pay or social security she can get by, you forget her medical benefits are not going to be of any use to her in Japan; the older someone gets, the more healthcare they will need.  Getting old is not just a matter of looking older, but your body starts to fail and need more and more medical care with each passing year. And not only that, Japan's healthcare system is already in crisis because the nation is aging with a near zero birthrate which means there are fewer young people and wage earners who can be taxed to pay for the care of the elderly. So even if your mother's retirement benefits are sufficient to live on, the cost of medical care she will have to pay out of her pocket is going to be substantial.

You might be able to qualify for a visa, if you meet the requirements. 

You wouldn't be able to take your mother with you.

Japan will sink so come to South Korea. I'll treat you to kimchi.

Retired people can often move to other countries and settle there, since they are not going to be looking for a job. Look on the website of the Japanese consulate, or talk to someone at a consulate. If you're in a fairly large city, there will be other expats for your mother to hang out with. But it's expensive in Japan, and the jobs for English teachers usually require at least a certificate in TESL/TEFL, preferably a master's degree. 

My suggestion is to consult the Japanese government's website and see what you need to do to qualify for this plan. Googling 'immigrate to japan' will find not only the official website but others dealing with the same subject.

You may not get accepted there. Your mom will be quite lonely in Japan. Please find other options.